Blindsight: very interesting question.
I think the way neurologists are viewed by other medical specialists has varied widely over time, and today still varies to a smaller extent on place of practice.
In the old days (ie. late 19th century until 1960s-70s) neurologists were generally been viewed as highly intellectual, if commonly ineffectual, colleagues.
These were the people you turned to when faced with any patient with any sort of problem anywhere in the nervous system that you couldn't make head or tail of. This was what would often happen:
He (and it was almost always a he) would turn up, sometimes after an unconscionably long period of time, proceed to politely take a thorough history, do an extremely detailed (and, if you bothered to watch him do it, sublimely elegant) examination, make subtle observations about all kinds of things no other medical consultant would have noticed if they had watched the patient twice as long, chuckle briefly, nod his head sagely, tell you the patient had something you'd never heard of in your entire life - named after 5 European guys you swear no one else anywhere had ever heard of in their entire lives - and to prove it, suggest a test (yes, just one) you didn't quite think was as important to get quite so soon.
Days later the test would come back positive (but of course), and you'd happily tell the neurologist when you next saw him in the hallway that he was right. He'd be somewhat surprised you felt this was news, but just thanks you - politely - once more for the "very interesting" consult.
He is about to walk on, when you tug at his sleeve and say "Umm, so what're you going to do?" He looks genuinely surprised at first that there should be something more to do, but then looks at you with grave eyes and mutters something about "outcomes not being very promising" in "these cases". If you'd "send her down to my clinic when she's discharged, I'd be happy to follow her along, of course".
That evening, long after everyone else had left, you see him chug off in his ancient Ford, after enough tinkering about in his lab for the day.
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Or something like that. And so the legend grew about the neurologist not being able to able to do anything much, though he certainly knew an incredible amount of medicine.
These days, many things have changed. But some things have still remained the same.
Neurologists can do a little more these days. A little clot busting here, a little immunosuppresion there. A few more functions preserved for their patients, a few more months or years of life. With luck, maybe fewer of those years are spent in a nursing home. Better pain control.
But so much still cannot be cured. A lot of other specialists unfairly (and stupidly) seem to think that this is somehow neurologists "fault". That they can't be of much use because they can't seem to really do anything.
This sort of thinking ignores a couple of important things.
Firstly, there's a whole bunch of diseases out there - in all specialties - that you just don't have any chance (currently) of curing. But not being able to cure does not mean not being able to help. To help a dying or suffering individual to any small extent is not a bad thing, and patients appreciate that.
Second, neurologic disease is usually bad stuff, no matter how you look at it. Having a smart guy figure out that you really do have an incurable problem and not a rare but treatable disease is also not a bad thing.
Thirdly, the reason so much of neurology is incurable has to do with the brain (duh). Of all things in medicine, the brain is what we understand the least. We don't understand even the very basics of how the brain functions, let alone the etiopathology of its diseases. Every other organ (and every other specialist) deals with an infinitely easier system. You can understand very easily how the heart works, from the physics to the chemistry, to a lot of the molecular biology. The thing is basically a pump. You can understand how the kidney works, from the molecular biology through the cellular and tissue level to the systems level. The thing is a filter and a fluid regulator.
But can you tell me how the brain works? How does a bunch of neurons produce your thoughts? your sense of self as a person, your behavior, your memories, emotions? How does your brain process pain? The other sensations? Heck, how does a bunch of neurons produce you?
How are these cells made? Are they renewed continuously, like the skin and gut? What controls them? If you could renew them, would they form their own circuits? What regulates those? What about transmitters?
We don't know BASIC stuff about the brain, how it works, how it can go wrong, how to repair it when it does. And this is not for lack of trying. It's just that the brain is just really hard (duh).
But with the incredible research going on, this is changing everyday. In the meantime, neurologists do their best. Sometimes however, that doesn't amount to much, from other specialists point of view.
It's interesting that you bring up the part I quoted above... these are exactly the types of questions I'm interested in researching. However, I'm not so sure that neurology will help me answer any of these things. I'm constantly wondering if I should have gone to graduate school in neuroscience or neurobiology. 
